List coloring hypergraphs

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1 List coloring hypergraphs Penny Haxell Jacques Verstraete Department of Combinatorics and Optimization University of Waterloo Waterloo, Ontario, Canada Department of Mathematics University of California San Diego, CA Submitted: Apr 29, 2010; Accepted: Sep 6, 2010; Published: Sep 22, 2010 Mathematics Subject Classification: 05C15, 05C65 Abstract Let H be a hypergraph and let L v : v V H) be sets; we refer to these sets as lists and their elements as colors. A list coloring of H is an assignment of a color from L v to each v V H) in such a way that every edge of H contains a pair of vertices of different colors. The hypergraph H is k-list-colorable if it has a list coloring from any collection of lists of size k. The list chromatic number of H is the minimum k such that H is k-list-colorable. In this paper we prove that every d-regular three-uniform linear hypergraph has list chromatic number at least log d 5log log d )1/2 provided d is large enough. On the other hand there exist d-regular three-uniform linear hypergraphs with list chromatic number at most log 3 d + 3. This leaves the question open as to the existence of such hypergraphs with list chromatic number olog d) as d. 1 Introduction A hypergraph H is k-uniform if every edge of H has size k, and d-regular if every vertex of H is in exactly d edges of H. A hypergraph is linear if any pair of distinct edges of the hypergraph intersect in at most one vertex. Let H be a hypergraph and let L v : v V H) be sets; we refer to these sets as lists. A list coloring of H is an assignment of an element of L v to each v V H) in such a way that every edge of H contains a pair of vertices assigned different elements. The hypergraph H is called k-list-colorable if it has a list coloring from any collection of lists of size k. The list chromatic number χ l H) of H, also called the choice number of H, is the minimum k such that H is k-list-colorable. In this paper, we study the list chromatic number of linear uniform regular hypergraphs. Partially supported by NSERC Research supported by an Alfred P. Sloan Research Fellowship and NSF Grant DMS the electronic journal of combinatorics ), #R129 1

2 1.1 List coloring graphs The notion of list-coloring is a generalization of the notion of proper coloring, and has been studied extensively for graphs. In particular, Alon [1] showed that every bipartite graph of minimum degree at least d has list chromatic number at least 1 log 2 2 d, improving log d log log d a preceding lower bound of order as d in [2]. On the other hand it is known [9] that the complete bipartite graph K d,d with d vertices in each part satisfies χk d,d ) = 1 + o1)) log 2 d. It has been asked whether every bipartite graph with maximum degree d has list chromatic number Olog d) see Alon and Krivelevich [4]) but this tantalizing problem remains open. Here and in what follows, the logarithm is taken to be the natural logarithm unless a base is explicitly displayed. 1.2 List coloring hypergraphs There seem to be very few results on list colorings of hypergraphs. Perhaps the most famous question on colorings of hypergraphs is the Erdős-Faber-Lovász conjecture see [6]): given an n-uniform linear hypergraph consisting of n edges, there is vertex-coloring of the hypergraph for which every edge receives all n colors. Equivalently, the conjecture states that any graph comprising the union of n edge-disjoint cliques of size n has chromatic number n. Via a more general result on coloring linear hypergraphs, Kahn [11] showed that that an n-uniform linear hypergraph can be vertex-colored with n + on) colors in such a way that the vertices in each edge all receive different colors. Recently, a simple proof of the Erdős-Faber-Lovász conjecture was announced. For Steiner triple systems with n vertices three-uniform hypergraphs in which every pair of vertices is covered exactly once a lower bound of order log n/ log log n for the list chromatic number was shown in [10]. In this paper, we concentrate on giving bounds on the list chromatic number of all linear regular three-uniform hypergraphs, which we refer to as triple systems. We shall check in this paper that the complete r-partite r-uniform hypergraph K r n with parts of size n has list chromatic number asymptotic to log r n as n. 1.3 Main Theorem The main result of this paper shows that every d-regular linear triple system has large list chromatic number: Theorem 1 There exists a constant d 0 such that for d d 0, every d-regular linear triple system H has log d ) 1/2. χ l H) > 5 log log d On the other hand, there exists a d-regular linear triple system with list chromatic number at most log 3 d + 3. The requirement of linearity in this theorem is necessary. Consider the triple system consisting of vertex set V = V 1 V n W with V i = 2 and W = n and where the electronic journal of combinatorics ), #R129 2

3 the edge set E consists of {e V : e W = 1, i : V i e}. This is an n-regular triple system whose list chromatic number is two, since for any assignment of lists we can choose different colors for the two vertices inside each V i. Relative to the theorem above, the most relevant open question is the existence of d-regular linear triple systems with list chromatic number olog d) as d. We leave this as an open problem. Problem. Do there exist three-uniform d-regular linear hypergraphs of list chromatic number olog d) as d? 2 List coloring complete r-partite hypergraphs In this section, we show that for infinitely many d, there exists a linear d-regular triple system with list chromatic number at most log 3 d+3. The example is the following threepartite triple system K 3 d L): if L is any d d latin square, then we create a linear triple system with three parts as follows. We index the rows of L by a set R of size d, the columns by a set C of size d, and then let R, C, [d] denote the parts of K 3 d L). Then the edges of K 3 d L) are all triples {r, c, i} with r R, c C, and such that i is in position r, c) in L. Note that by definition of a latin square there are no repeated entries in any row or column of L, so the triple system K 3 d L) is d-regular and linear. We will show that it has list chromatic number at most log 3 d + 3 by showing that the complete r-partite hypergraph K r d which contains K 3 d L) when r = 3 has list chromatic number at most log r d + 3. We will also show that K r d has list chromatic number at least log r d Olog log d) as d. Our argument is based on that of [9]. We define a hypergraph H to be r-colorable if the list assignment in which all lists are {1,...,r} admits a list coloring of H. The following lemma will be used: Lemma 2 Let m r k) denote the minimum number of edges in a k-uniform hypergraph which is not r-colorable. Then as k, r k 1 m r k) O r k 2 r k ). Proof A random r-coloring of a hypergraph consists in choosing uniformly at random a color from [r] independently for each vertex in the hypergraph. The lower bound of r k 1 in the lemma follows from the fact that in a random r-coloring of a k-uniform hypergraph with m edges, the expected number of edges whose vertices are all assigned the same color is r 1 k m, so if m < r k 1, then the hypergraph is r-colorable. In fact, this is all we shall need to prove χ l K r n ) log r n + 3. The upper bound on m r k) also follows from probabilistic methods; we sketch the proof for r > 2, since it is very similar to the proof for r = 2 given in [7] see also Alon and Spencer [5], Page 9). We consider a hypergraph H with vertex set [rk 2 ] constructed by randomly and uniformly selecting sets of size k in T independent rounds from [rk 2 ]. We shall take T = 2k 2 er) k log r. the electronic journal of combinatorics ), #R129 3

4 An r-coloring of H is a partition X 1, X 2,...,X r ) of the vertex set [rk 2 ] of H. An edge of H is monochromatic if it lies entirely in some X i. We say that an r-coloring of H fails in round i if in the ith round, the chosen set of size k is monochromatic. The aim is to show that the expected number of r-colorings which do not fail in any round is less than one, and we follow the computation of Erdős [7]. The probability that the edge chosen at round i is monochromatic for a given coloring X 1, X 2,...,X r ) is exactly ) rk 2 1 r ) Xi. k k i=1 By convexity of binomial coefficients, the sum is a minimum when X 1 = X 2 = = X r = k 2 : r ) ) Xi k 2 r. k k i=1 Using the standard bounds ) rk 2 k erk) k and ) k 2 k k k, the probability that an edge chosen at round i is monochromatic is at least r/er) k. It follows that the probability that a given coloring X 1, X 2,...,X r ) does not fail at any round is at most 1 r ) T er) k since the rounds are independent. Now T was chosen so that this quantity is less than r rk2. Since the number of r-colorings of [rk 2 ] is r rk2, we conclude that the expected number of r-colorings which do not fail at any round is less than one. In particular, there is a witness H to this event. This hypergraph H has exactly T edges and is not r-colorable, as required. We use the bounds on m r k) in the above lemma to give bounds on χ l K r m ) as follows: Theorem 3 For all r 2 and as m, χ l K r m ) = 1 + o1)) log r m. Proof If we can show χ l K r m ) k when rm = m r k) 1 and χ l K r m ) k + 1 when m = m r k), then we are done using the bounds in the last lemma, since rm+1 r k 1 implies k log r m + 3 and m Ck 2 r k implies k log r m 2 log r log r m O1) when m, as required. First we show that for m = m r k), we have χ l K r m ) > k. By definition of m r k) = m, there is a k-uniform hypergraph F with m edges such that F is not r-colorable. We let the lists in the ith part V i of K r m be exactly the edges of F, for 1 i r. We claim there is no coloring of K r m from this list assignment. Suppose, for a contradiction, that there is a list coloring of K r m from these lists, and let S 1, S 2,..., S r be the sets of colors used on the vertices of V 1, V 2,...,V r, respectively. Note that each S i is actually a set of vertices of F. We observe that r S i = i=1 the electronic journal of combinatorics ), #R129 4

5 otherwise a certain color appears on a vertex v i V i for 1 i r, in which case {v 1, v 2,...,v r } is a monochromatic edge of K r m. Since each S i is a transversal of the edges in F, T i = V F)\S i does not contain any edges of F for 1 i r. Since S i =, every vertex of F is in the complement of some S i and therefore T 1 T r = V F). This contradicts that F is not r-colorable, and hence χ l K r,m ) > k when m = m r k). Conversely we show χ l K r m ) k when rm = m r k) 1. Consider an assignment of lists of size k to the rm vertices of K r m, and let F be the k-uniform hypergraph of all those lists. Then F has an r-coloring since F < m r k). Fix an r-coloring of F, say V F) = T 1 T 2 T r. The list L v at a vertex v V i must contain an element not in T i, otherwise L v would be monochromatic under the coloring T 1 T 2 T r of F. To each v V i we assign an arbitrary element of L v \T i, for 1 i r. We claim this is a proper coloring of K r m. If not, then there is an edge {v 1, v 2,...,v r } in K r m where v i V i, and such that every v i : 1 i r receives the same color, which is an element of T j for some j [r]. However, v j was assigned an element of L vj \T j, which is a contradiction. Therefore χ l K r m ) k. This completes the proof. It appears to be an interesting question to determine fd) = min L χ l K 3 d L)) where the minimum is over all d d latin squares. In particular, it would be interesting to determine whether fd) = olog d) as d. 3 Lemmas The probabilistic lemmas we use to prove Theorem 1 are given here. The first lemma is the Chernoff bound, one of the basic tools in probabilistic methods see for example Alon and Spencer [5]. Lemma 4 Let Z 1, Z 2,...,Z n be identically distributed independent random variables where PZ i = 1) = p and PZ i = 0) = 1 p, and let S be their sum. Then for any ǫ 0, 1], P S ES) > ǫes)) 2 exp ǫ 2 ES)/2). We also require the Lovász Local Lemma [8] in the following form. Here and in what follows, A c denotes the complement of an event A. Lemma 5 Let A 1, A 2,...,A n be events in a probability space, such that each A i is mutually independent of any subset of events indexed by the set [n]\j i for some dependency set J i [n] where max J i =. Suppose that for all i [n]. Then n P i=1 PA i ) 1 4 A c i) exp n ). the electronic journal of combinatorics ), #R129 5

6 4 Proof of Theorem 1 We are given a d-regular linear triple system H, and we want to come up with a collection of lists on the vertices from which no coloring is possible and where the lists are as large as possible. The idea of the proof is to show that if we assign random lists of length s in [t] := {1, 2,..., t} to the vertices of H, for some carefully chosen values of s and t, then with positive probability there is no proper coloring of H from these lists. We will choose s = log d) 1/2 5 log log d) 1/2 and t = 8s) ss+4) and put p := 1/8s) 3 t. If d is a large enough constant, then it is straightforward to check that the parameters p, s, t satisfy for completeness a verification is written in the appendix): t exp s 2 p 3 d/4t 2 ) < 1/64d 2 1) exp n/8s) 2 t) < s 3pn 2) exp p 2 n/8) < exp n/d 2 ). 3) For any K 8s) s, we record that standard bounds on binomial coefficients see Appendix : Lemma 6) give ) ) t/k t > 2K) s. 4) s s 4.1 Preprocessing Define random sets X V H) and Y V H) by independently placing each vertex of V H) into X with probability p 2 and into Y with probability p and into Z = V H)\X Y ) with probability 1 p p 2. Let H denote the three-partite hypergraph consisting of all {x, y, z} H such that x X, y Y and z Z. We write dx, y) > 2 to denote that in H every edge on x is disjoint from every edge on y in other words x and y are at distance more than two. For y Y let Γ X y) = {x X : {x, y, z} H for some z} and for z Z let Γ XY z) = {x, y) X Y : {x, y, z} H }. Define d X y) = Γ X y) and d XY z) = Γ XY z). Since H is a d-regular linear hypergraph, we have for y Y and z Z, Ed X y)) = 2p 2 1 p p 2 )d and Ed XY z)) = 2p 3 d. Let A y : y Y be the events p 2 d < d X y) < 4p 2 d and let A z : z Z be the event p 3 d < d XY z) < 4p 3 d. Claim 1. With positive probability, each event A X = { 1 2 p2 n < X < 2p 2 n} and A Y = { 1 2 pn < Y < 2pn} and every A y : y Y and every A z : z Z occurs. Proof. We shall apply the Chernoff Bound with ǫ = 1/ 8. The event A y contains the event d X y) Ed X y)) < ǫed X y)), since p < 1/8 3, and clearly A z contains the event the electronic journal of combinatorics ), #R129 6

7 d XY z) Ed XY z)) < ǫed XY z)). Since vertices are placed independently in the sets X and Y, Lemma 4 shows PA c y) 2 exp p2 d ) 8 and PA c z) 2 exp p3 d ) 8 and similarly PA c X ) 2 exp p2 n/8) and PA c y ) 2 exp pn/8). A dependency graph of the events A y : y Y and A z : z Z has maximum degree at most = 4d 2, since any single event A c v is mutually independent of any set of events Ac w : dv, w) > 2. By 1), both the bounds in 5) are easily less than 1/4. By Lemma 5, with probability at least exp n/4d 2 ) every A y : y Y and every A z : z Z occurs. By 3), exp n/4d 2 ) > PA c X Ac Y ), and so with positive probability, the events A X and A Y and every A y : y Y and A z : z Z occurs. This proves the claim. For the remainder of the proof, we work in a subhypergraph H for which all the events in Claim 1 hold, and we assume that X, Y and Z are the parts of H. 4.2 Choice of lists in X First we assign lists to X. We choose uniformly and independently random lists of size s from [t] for the vertices of X. For y Y, let B y be the event that no color appears in more than 2sd X y)/t lists in Γ X y) and for z Z let B z be the event that no color appears in more than 2sd XY z)/t lists in Γ X z) := {x X : y Y, x, y) Γ XY z)}. 5) Claim 2. With positive probability, every B y : y Y and every B z : z Z occurs. Proof. Since H is linear, note that Γ X z) = d XY z). The expected number of times a particular color appears in lists in Γ X y) is exactly sd X y)/t and the expected number of times a particular color appears in Γ X z) is sd XY z)/t. Since the lists are chosen independently, the probability that a particular color appears in more than 2sd X y)/t lists in Γ X y) is at most 2 exp sd X y)/2t), by the Chernoff Bound with ǫ = 1. A similar statement holds for the colors in Γ X z), and since there are at most t colors, we deduce from the union bound that PB c y ) < 2t exp sd Xy)/2t) < 2t exp sp 2 d/2t) PB c z ) < 2t exp sd XY z)/2t) < 2t exp sp 3 d/2t). A dependency graph of the events Bv c has maximum degree at most = 4d2, since Bv c is mutually independent of any events Bw c : dv, w) > 2. By 1), we easily have PBy c) < 1/4 and PBc z ) < 1/4, and so Lemma 5 completes the proof of Claim 2. From now on we fix an assignment of lists L of size s from [t] to the vertices of X, such that every B y : y Y and every B z : z Z occurs. Let ρ y be the number of colors the electronic journal of combinatorics ), #R129 7

8 used at least d X y)/2t times on Γ X y). Since B y occurs, t ρ y ) d Xy) 2t + ρ y 2sd X y) t d X y) and it follows that ρ y > t/4s. Since every B y occurs, we have that for each y Y and each coloring of X from L there exists a set S y of t/4s colors each appearing at least p 2 d/2t times in Γ X y). 4.3 Choice of lists in Y Now we independently and randomly assign lists L y of size s from [t] to the y Y. For a fixed coloring of X from L, if L y S y, then any color selected from L y results in at least p 2 d/2t vertices x Γ X y) of the same color as y. Let B χ be the event that under a coloring χ of X, L y S y for at least 1 2 8s) s Y vertices y Y, and let B = χ B χ where the intersection is over all colorings of X from L. For z Z, let C z be the event that no color appears on the list at x and at y for at least 4s 2 d XY z)/t 2 pairs x, y) Γ XY z). Claim 3. With positive probability, B as well as every C z : z Z occurs. Proof. We observe that for every χ and every y, ) ) t/4s t PL y S y ) / > 8s) s s s using 4) with K = 4s. Therefore the expected number of events L y S y is at least 8s) s Y. Since the L y are chosen independently, the Chernoff Bound with ǫ = 1/2 shows that for a fixed coloring χ of X, PBχ c ) < 2 exp Y ). 88s) s Since there are at most s X colorings of X, and since by Claim 1 Y pn/2 and X 2p 2 n, the expected number of colorings χ of X for which Bχ c occurs is at most 2 exp Y ) s X < 2 exp pn ) 88s) s 168s) + s 2p2 n log s < exp pn ). 208s) s By 2), with room to spare, this is less than exp n/d 2 ), and so Markov s Inequality gives PB c ) < exp n/2d 2 ). For x Γ X z), there is a unique y Γ Y z) such that x, y) Γ XY z), since H is linear. The chance that y is assigned a list containing a particular color in L x is exactly s/t. Fix a color i, and let C i z be the event that color i appears on the list at x and at y for at least 4s 2 d XY z)/t 2 pairs x, y) Γ XY z). Recall that since every B z occurs, color i appears at most 2sd XY z)/t times in Γ X z). The expected number of x, y) Γ XY z) such that i L x L y is at most 2s 2 d XY z)/t 2. Since lists are assigned independently, we may apply the Chernoff Bound with ǫ = 1 to obtain PC i z ) < 2 exp s2 d XY z)/t 2 ). the electronic journal of combinatorics ), #R129 8

9 Since there are t colors in total, the union bound shows By Claim 1, this is at most PC c z ) < 2t exp s2 d XY z)/t 2 ). t exp s 2 p 3 d/2t 2 ). Now Cz c is mutually independent of any events Cc w : dw, z) > 2, so a dependency graph of events Cz c has maximum degree at most = 4d 2. By 1), PCz) c < 1/4 and so Lemma 5 shows that with probability at least exp n/4d 2 ), every C z occurs. Using the preceding bound on PB c ), we see that with positive probability every C z : z Z and B occurs. Let us now extend our list assignment L to include a list assignment of s elements from [t] to each y Y, such that B and each C z : z Z occurs. 4.4 Choice of lists for Z We have assigned lists L to the vertices of X and Y in such a way that in every coloring of X, there is a set S Y of size at least pn/48s) s with the property that for y S, L y S y this is the event B), and for every z Z, every color appears on the list at x and at y for at most 4s 2 d XY z)/t 2 pairs x, y) Γ XY z) these are the events C z ). Now we show that there exists an assignment of lists to Z from which no coloring of H is possible. Let G denote the bipartite graph consisting of parts X and Y and edges z Z Γ XY z). If κ is any fixed coloring of X Y, then since B occurs, whichever color from L y that is assigned to y by κ results in at least p 2 d/2t pairs x, y) G such that x and y have the same color. Therefore we have at least p 2 d 2t > p3 dn 8t8s) s y S pairs x, y) G such that x and y are assigned the same color by κ. We call these monochromatic edges of G. Let T denote the set of z Z such that Γ XY z) contains at least p 3 d/t8s) s+1 monochromatic edges under κ. Since every C z occurs, no Γ XY z) can contain more than t 4s 2 d XY z)/t 2 16s 2 p 3 d/t monochromatic edges under κ. Therefore T 16s2 p 3 d t + n T ) p 3 d t8s) p3 dn s+1 4t8s) s. From this it follows that T > n/8s) s+2. Since every C z occurs, for every z T, Γ XY z) contains monochromatic edges of at least p 3 d t8s) s+1 t 2 16s 2 p 3 d > 2t 8s) s+3 different colors under κ. Let T z,κ be this set of colors on monochromatic edges of Γ XY z). We assign random lists to Z of size s from [t]. In order for H, and therefore H, to have the electronic journal of combinatorics ), #R129 9

10 a list coloring extending κ, it cannot be that some list at a vertex z T is contained in the set T z,κ. Let C κ be the event that no list at any vertex z T is contained in T z,κ. By 4) with K = 8s) s+3 /2, ) 2t/8s) s+3 and therefore PC κ ) < 1 2t/8s) s+3 s t s) s > 1 2K) s ) t s )) T < exp T ) < exp n ) 2K) s 8s) 2 t using that T > n/8s) s+2. By 2), PC κ ) < s 3pn, and since there are at most s X Y s 3pn colorings κ of X Y, we deduce that there exists a list assignment to the vertices of Z for which no C κ occurs. In words, there is an assignment of lists to the vertices of Z such that for any coloring κ of X Y, some z Z cannot be properly colored from its list. For this list assignment, no coloring of H exists. This completes the proof. 5 Note added in proof We have learned that the list coloring problem for linear and uniform hypergraphs has been studied independently by Alon and Kostochka [3]. They prove a more general version of Theorem 1, in that they consider r-uniform hypergraphs for r 3, and they do not restrict to d-regular hypergraphs but give bounds in terms of the average degree d. When specialised to the setting of Theorem 1, their results give similar bounds, and in particular we still do not know whether there exist 3-uniform d-regular hypergraphs with list chromatic number olog d) as d. 6 Appendix We chose s = log d) 1/2 5 log log d) 1/2 and t = 8s) ss+4) and put p := 1/8s) 3 t. If d is a large enough constant, we claim that the following three inequalities hold: t exp s 2 p 3 d/4t 2 ) < 1/64d 2 6) exp n/8s) 2 t) < s 3pn 7) exp p 2 n/8) < exp n/d 2 ). 8) To verify the inequalities 6), 7), 8) we first observe that with the given definitions we have s 2s2 < d 1/5 and so s s2 < d 1/10. Thus when d is large enough we find t = 8s) ss+4) < d 1/9. Therefore to prove 6) we note that for large enough d, t exp s 2 p 3 d/4t 2 ) < d 1/9 exp s 2 d/48s) 9 t 5 ) < d 1/9 exp d 1/3 ) < 1/64d 2. For 7) it suffices to show 3p log s < 1/64s 2 t, which is immediate from the definition of p. Finally for 8) we want 1/d 2 < p 2 /8, so since p 2 = 1/8s) 6 t 2 > 1/d 1/3 the inequality holds provided d 5/3 > 8. the electronic journal of combinatorics ), #R129 10

11 Lemma 6 Let s, t be positive integers and suppose that t/k 2s. Then ) ) t/k t > 2K) s. s s Proof By definition we have ) t/k > s t/k s)s s! > 1 s! t ) s. 2K On the other hand and the result follows. ) t ts s s! Acknowledgments This work was done while the authors were visiting the Institute for Pure and Applied Mathematics at the University of California, Los Angeles. The authors wish to thank IPAM for their support. References [1] Alon, N., Degrees and choice numbers, Random Structures & Algorithms ), [2] Alon, N., Restricted colorings of graphs. In: K. Walker, Editor, Surveys in Combinatorics, Proc 14th British Combinatorial Conference, London Mathematical Society Lecture Notes Series 187, Cambridge Univ. Press 1993), [3] Alon, N., Kostochka, A, Hypergraph list coloring and Euclidean Ramsey theory, submitted, [4] Alon, N., Krivelevich, M., The choice number of random bipartite graphs, Annals of Combinatorics ), [5] Alon, N., Spencer, J. The Probabilistic Method, Wiley-Interscience, [6] Erdős, P., On the combinatorial problems I would most like to see solved, Combinatorica ), [7] Erdős, P., On a combinatorial problem. II. Acta Mathematica Academiae Scientiarum Hungaricae ), [8] Erdős, P., Lovász, L., Problems and results on 3-chromatic hypergraphs and some related questions, in Infinite and Finite Sets A. Hajnal et al, eds.), North-Holland, Amsterdam 1975), the electronic journal of combinatorics ), #R129 11

12 [9] Erdős, P., Rubin, A., Taylor, H., Choosability in graphs, Proc. West Coast Conf. on Combinatorics, Graph Theory and Computing, Congressus Numerantium ), [10] Haxell, P., Pei, M., On list coloring Steiner triple systems, Journal of Combinatorial Designs ), [11] Kahn, J., Coloring nearly-disjoint hypergraphs with n + on) colors, Journal of Combinatorial Theory A ), the electronic journal of combinatorics ), #R129 12